Subscribe to this blog

Follow by Email

Search This Blog

Divisibility by Nine Number Trick

A well known divisibility trick exists to tell if a number is divisible by 9. First add all the digits of that number and if the result is divisible by 9, then that number is also divisible by 9. How does one prove this?

For convenience let us assume it is a 4 digit number we are testing, say \(abcd\). This can be expressed as
$$
\text{abcd} = 1d + 10c + 100b + 1000a
$$
This in turn can be expressed as
$$
d + c + b + a + 9c + 99b + 999a \\
\text{or}\\
d + c + b + a + 9\times(c + 11b + 111a)\\
$$
Note that the second part of the number \(9(c + 11b + 111a)\) is divisible by 9. If \(d+c+b+a\) is divisible by 9, then the whole number is divisible by 9. Hence the proof. By extension, as 9 is divisible by 3, the method applies to 3 as well.
The following are some good books on learning probability

This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"

This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.

This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.